What's the difference between operating deflection shapes and mode shapes? 

Sometimes they look the same to me!

Well . . . let's describe the differences.

工作变形和模态振型之间有什么差别？

对我来说，有时看起来它们像是一回事！

好吧… 那我们说说其差别吧。

This is a common stumbling point for many people. This is partly due to the words that we use. I would much rather call the data we receive from an operating condition, an operating deflection pattern, rather than use the word shape. But unfortunately, I can't change the nomenclature at this point.

对很多人来讲，这是一个常见的容易出错的问题。一部分原因是由我们使用的这些词语造成的。我倒是宁愿把工作条件下采集的数据称为工作变形模式，而不愿意使用形式这个词。但不幸的是，目前我不能改变这个专门术语了。

 

Let's first recall how a structure responds, in general, due to any excitation

让我们首先回想一下结构是如何响应的，通常是由任意激励引起的

Of course, we realize that the input forcing function is actually applied in the time domain but we represent it in the frequency domain; also the response actually occurs in the time domain but it can also be represented it in the frequency domain.

我们当然知道，输入激励函数实际上是施加在时域，但是我们却在频域内表示它；同样，响应也发生在时域内，但是也同样可以在频域内表示。

So for a structure which is exposed to an arbitrary input excitation, the response can be computed using the frequency response function multiplied by the input forcing function. This is very simply shown in the schematic in Figure 1.

因此，对于一个受到任意输入激励的结构，可以利用频响函数乘以输入激励函数来计算响应。在图1的示意图中，对此进行了非常简单的说明。

 

The excitation shown is a random excitation that excites all frequencies. The most important thing to note is that the frequency response function acts as a filter on the input force which results in some output response. The excitation shown causes all the modes to be activated and therefore, the response is, in general, the linear superposition of all the modes that are activated by the input excitation. Now what would happen if the excitation did not contain all frequencies but rather only excited one particular frequency (which is normally what we are concerned about when evaluating operating conditions).

所示的激励为随机激励，对所有频率进行激发。最重要的一点要注意，频响函数作为滤波器，作用在输入激振力上，这样得到了某种输出响应。所示的激励激起了所有阶模态，因此就一般情况下，响应是输入激励激发起来的所有阶模态的线性叠加。现在，如果激励不包含所有频率而只是激起了一个特定的频率（评价工作状态时，我们所关心的通常是此类情况），情况又会怎样。

 

Let's consider a simple plate that is excited by an input force that is sinusoidal in nature. And let's also assume that the force is applied at one corner of the plate. For the example here, we are only going to consider the response of the plate assuming that there are only 2 modes that are activated by the input excitation. (Of course there are more modes, but let's keep it simple to start.) Now from figure 1 and equation 1 we realize that the key to determining the response is the FRF between the input and output locations. Also, we need to remember that when we collect operating data, we don't measure the input force on the system and we don't measure the system FRF – we only measure the response of the system.

让我们来考虑一个简单平板，平板受到按正弦方式变化的输入激振力的激励。而且也假设激振力施加于平板的某个角上。对于本例，将只考虑平板的响应，假设输入激励仅仅激起了2阶模态（当然还有更多阶的模态，但让我们从简单入手）。现在，根据图1和公式1，我们了解到，决定响应的关键在于输入和输出位置之间的FRF。同样，需要记住，当采集工作数据时，我们不去测量作用到系统上的输入激振力，也不测量系统FRF — 我们仅仅测量系统响应。

 

First let's excite the system with a sinusoid that is right at the first natural frequency of the plate structure. The response of the system for one FRF is shown in Figure 2. So even though we excite the system at only one frequency, we know that the FRF is the filter that determines how the structure will respond. We can see that the FRF is made up of a contribution of both mode 1 and mode 2. We can also see that the majority of the response, whether it is in the time or frequency domain, is dominated by mode 1. Now if we were to measure the response only at that one frequency and measure the responses at many points on the structure, then the operating deflection pattern would look very much like mode 1 - but there is a small contribution due to mode 2. Remember that with operating data, we never measure the input force or the FRF - we only measure the output response. So that the deformations that are measured are the actual response of the structure due to the input excitation - whatever it may be.

首先，我们用一个正弦信号对系统进行激励，信号频率正好等于平板结构第1阶固有频率。对某个FRF，系统响应如图2所示。那么，尽管仅在一个频率上对系统进行激励，但我们知道，FRF是滤波器，决定了结构会如何响应。可以看出，FRF由1阶模态和2阶模态两者的贡献共同组成。同时也看出，大部分响应，不管是在时域内还是在频域内，是由模态1占主导的。现在如果我们去测量那个频率下的响应，而且是测量结构上很多点的响应，那么看起来，工作变形模式将与模态1非常类似 — 但有少量的2阶模态引起的贡献。记住，对于工作数据，我们从来不去测量输入激振力或FRF — 我们仅仅测量输出响应。这样一来，测得的变形是输入激励引起的结构的实际响应 — 而不论输入激励是什么。

When we measure FRFs and estimate modal parameters, we actually determine the contribution to the total FRF solely due to the effects of mode 1 acting alone, as shown in blue, and mode 2 acting alone, as shown in red, and so on for all the other modes of the system. Notice that with operating data, we only look at the response of the structure at one particular frequency - which is the linear combination of all the modes that contribute to the total response of the system. So we can now see that the operating deflection pattern will look very much like the first mode shape if the excitation primarily excites mode one.

当我们测量FRFs进而估计模态参数时，实际上是确定1阶模态单独作用的效应对于总体FRF的独立贡献，如蓝色所示；2阶模态单独作用，如红色所示；同时，对于所有其他阶的系统模态，可依此类推。注意，对于工作数据，我们仅仅是观察结构在某个特定频率下的响应 — 它是对系统总体响应有所贡献的所有阶模态的线性合成。因此我们现在可以看出，如果激励主要激起了1阶模态，那么看起来工作变形模式与第1阶模态振型将非常类似。

 

Now let's excite the system right at the second natural frequency. Figure 3 shows the same information as just discussed for mode 1. But now we see that we primarily excite the second mode of the system.  Again, we must realize that the response looks like mode 2 - but there is a small contribution due to mode 1.

现在，让我们正好在第2阶固有频率上对系统进行激励。图3显示出与刚才所讨论的1阶模态相同的信息。但是现在看到，主要是激起了系统的第2阶模态。我们必须再一次认识到，响应看起来跟2阶模态很像 — 但是存在1阶模态引起的少量贡献。

But what happens when we excite the system away from a resonant frequency. Let's excite the system at a frequency midway between mode 1 and mode 2. Now here is where we see the real difference between modal data and operating data. Figure 4 shows the deformation shape of the structure. At first glance, it appears that the deformation doesn't look like anything that we recognize. But if we look at the deformation pattern long enough, we can actually see a little bit of first bending and a little bit of first torsion in the deformation. So the operating data is primarily some combination of the first and second mode shapes. (Yes, there will actually be other modes but primarily mode 1 and 2 will be the major participants in the response of the system.)

但是，当我们远离固有频率对系统进行激励时，情况又会如何呢？让我们在1阶模态和2阶模态中间位置的一个频率上对系统进行激励。现在，可以看出模态数据和工作数据之间存在的实际差别在哪儿了。图4表示结构的变形形式。初看之下，这个变形看上去跟我们认识的任何东西都不像。但是，如果我们观察这个变形模式足够长的时间，我们真的可以看出在变形中有一点点一阶弯曲，还有一点点一阶扭转。所以，工作数据主要是1阶模态振型和2阶模态振型的某种合成。（是的，实际上还有其他阶模态，但是，1阶模态和2阶模态是系统响应中的主要参与者。）

Now, we have discussed all of this by understanding the FRF contribution on a mode by mode basis.  When we actually collect operating data, we don't collect FRFs but rather we collect output spectrums.  If we looked at those, it would not have been very clear as to why the operating data looked like mode shapes. Figure 5 shows a resulting output spectrum that would be measured at one location on the plate structure. Now the input applied to the structure is much broader in frequency and many modes are excited. But, by understanding how each of the modes contributes to the operating data, it is much easier to see how the modes all contribute to the total response of the system. So actually, there is a big difference between operating deflections and mode shapes - we can now see that the modes shapes are summed together in some linear fashion to form the operating deflection patterns. I hope that this helps to clear up the mystery as to the differences between operating deflection patterns and mode shapes.

目前，在模态基础上，通过理解某一阶模态的FRF贡献的方式，我们已经讨论完了这个问题的所有方面。当实际采集工作数据时，我们不去采集FRFs，而是采集输出谱。如果我们观察它们，一直都不是很清楚为什么工作数据看起来像是模态振型。图5显示了在平板结构的某一个位置进行测量而得到的输出谱。现在，作用到结构上的输入是频率上更为宽带的信号，而且激起了很多阶模态。但是，通过理解每一阶模态是怎样对工作数据有所贡献的，就很容易弄明白所有阶模态是怎样贡献于系统的总体响应了。所以实际上，工作变形和模态振型之间有着巨大的差别 — 现在我们可以看出，模态振型按某种线性方式一起求和，构成了工作变形模式。我希望这个讨论有助于澄清关于工作变形模式和模态振型之间的谜团。

Think about it and if you have any more questions about modal analysis, just ask me.

好好思考一下这个问题，如果你有关于模态分析的任何其他问题，尽管问我好了。